Time-domain & frequency-domain synthesis
Close the second-order course by showing that poles, ring-down, damping, step overshoot, resonance bandwidth and Q are different measurements of the same ideal two-state dynamics only when their assumptions match. Derive Δω=2α for the ideal series-RLC current response, Q=ω0/(2α), the ring-down envelope time 1/α, logarithmic decrement, independent time/frequency parameter extraction, and consistency checks that reveal loading, zeros, parasitics or model failure.
- Map the canonical poles s=−α±jωd to ω0, ζ, Q and the underdamped ring-down envelope.
- For the ideal series-RLC current/resistor response, prove Δω=2α and therefore Q=ω0/(2α).
- Derive the ring-down amplitude e-folding time τenv=1/α and its exact relation τenv=2/Δω for that series-RLC response.
- Express the bandwidth relation in hertz as Δf·τenv=1/π and explain its model scope.
- Derive logarithmic decrement δ=αTd=2πα/ωd from successive same-sign peaks of an underdamped natural response.
- Recover α, ωd, ω0, ζ and Q from a measured ideal ring-down using peak spacing and amplitude ratio.
- Recover ω0, Δω and Q independently from the half-power edges of the ideal series-RLC resistor/current frequency response.
- Invert the canonical step-overshoot relation to estimate ζ only when the declared all-pole unit-step assumptions are satisfied.
- Compare independently extracted time-domain and frequency-domain parameters and use disagreement as evidence of loading, zeros, extra states, nonlinearities or incorrect measurement references.
- Build a final decision map separating universal second-order pole facts from topology/output-specific resonance and bandwidth identities.
We have now seen the same ideal series RLC as a two-state differential equation, a pair of poles, a ring-down, a step response and a resonant frequency response. Which numerical parameters are truly the same across those views, and which relations depend on the chosen topology and output?
05.1The pole pair is the common core
For an underdamped second-order denominator,
and
For the canonical denominator
the pole facts are independent of which state or output is later plotted. Numerators, however, can change zeros, step shape and frequency-response peaks.
Figure 5.1 — One model, several observables
The one question is: which measurements should recover the same parameters if the ideal RLC model is valid?
All four paths should converge to the same parameters only when the same second-order LTI model, the same loading conditions and the correct output quantity are being used.
05.2Ring-down reveals α and ωd directly
Reference schematic — source-free series RLC
The source is suppressed; the i, vR, vL and vC references remain exactly those used in the equations.
A real underdamped natural response can be written
The oscillation repeats after
while the exponential envelope has e-folding time
These are independent pieces of information: Td measures the imaginary part of the poles, while the envelope measures their negative real part.
05.3Logarithmic decrement from successive same-sign peaks
Successive positive peaks of the damped sinusoid are separated by Td. Their magnitudes therefore satisfy
Define logarithmic decrement
Then exactly for the ideal damped sinusoid,
Using ζ=α/ω0 and ωd=ω0√(1−ζ²),
This provides a direct time-domain estimator of damping without needing a frequency sweep.
05.4Recover the full pole parameters from ring-down data
Measure two successive same-sign peaks separated by Td. Then
Recover
The Q relation here uses the conventional second-order denominator definition. It is not yet a statement about a particular half-power bandwidth; that bridge comes next for the declared series-RLC response.
05.5For the ideal series-RLC current response, bandwidth equals 2α
Reference schematic — driven series RLC
The arrows and polarities below define exactly the variables used in KVL.
For the series RLC, chapter 01 gave
Chapter 04 derived the exact half-power bandwidth of the current/resistor response:
Therefore
and
This is a powerful time/frequency equivalence, but its exact bandwidth meaning belongs to this topology, response and half-power criterion.
05.6Ring-down time and bandwidth are reciprocal here
Since τenv=1/α and Δω=2α,
Using Δω=2πΔf,
Also, since ω0=2πf0,
A narrow ideal series-RLC current resonance therefore corresponds to a long ring-down envelope. This is the same loss physics viewed in frequency and time.
05.7Frequency sweep gives an independent parameter set
For the ideal series-RLC resistor/current response, measure the two half-power angular frequencies. Then
The predicted natural ring-down frequency is then
A measured ring-down period provides an independent check of that prediction.
05.8Step overshoot provides a third check—but only for the canonical step
For the canonical stable all-pole underdamped unit-step response from zero initial state, chapter 03 derived
Let Lp=−ln Mp>0. Solving for ζ gives
Then Q=1/(2ζ) for the same canonical denominator. But if zeros, feedthrough or nonzero initial energy alter the waveform, this inversion no longer estimates damping reliably.
05.9Use disagreement as evidence, not as nuisance
Suppose ring-down gives one Q while a frequency sweep gives another. Possible causes include:
- source or load resistance changed the effective R;
- the measured output has zeros or a different resonance condition;
- parasitic C/L introduced additional states;
- component loss is frequency dependent;
- the circuit is nonlinear over one of the measurement amplitudes;
- the initial condition or half-power reference was extracted incorrectly;
- the measurement instrument loaded the circuit.
Do not average inconsistent Q values before explaining why they differ. Model validation is the purpose of the cross-domain comparison.
05.10Final mastery synthesis
- A ring-down has same-sign peaks 1.00 V and 0.60 V separated by 0.80 ms. Find δ, α, ωd, ω0, ζ and Q.
- For an ideal series RLC with half-power edges 9 krad/s and 11 krad/s, find ω0, Δω, α and Q.
- Using the result of exercise 2, predict τenv and ωd.
- Show algebraically that Δfτenv=1/π under the declared series-RLC assumptions.
- A canonical step has 10% overshoot. Estimate ζ and Q.
- Why can the Q from capacitor-voltage peak width differ from the simple resistor-output half-power Q if the criterion is not defined identically?
- Name four physical reasons time-domain and frequency-domain estimates can disagree.
- Classify each relation as pole-universal or series-RLC/output-specific: ω0²=α²+ωd²; Q=1/(2ζ); Δω=R/L; ωpeak,C=ω0√(1−2ζ²); δ=2πα/ωd.
Solutions
1. δ=ln(1/0.60)=0.51083. α=0.51083/0.0008≈638.53 s⁻¹. ωd=2π/0.0008≈7853.98 rad/s. ω0≈7879.89 rad/s. ζ≈0.08103 and Q≈6.17.
2. ω0=√(9×11) krad/s≈9.9499 krad/s. Δω=2 krad/s; α=1 krad/s; Q≈4.9749.
3. τenv=1 ms; ωd=√(ω0²−α²)≈9.8995 krad/s.
4. Δω=2α and Δf=Δω/(2π), while τenv=1/α.
5. Lp=−ln0.1=2.30259; ζ=Lp/√(π²+Lp²)≈0.5912; Q≈0.8458.
6. The capacitor transfer has a different magnitude shape and peak condition; bandwidth must be tied to a declared response and reference level.
7. Loading, extra parasitic states, frequency-dependent loss, nonlinear amplitude dependence, zeros/output choice, extraction/reference errors or instrument loading.
8. Pole/canonical-denominator relations: ω0²=α²+ωd², Q=1/(2ζ), δ=2πα/ωd. Series-RLC/output-specific: Δω=R/L and the capacitor-output peak expression.
DY-02 is now complete: two storage states became a second-order ODE, natural roots, damping regimes, forced step response, resonance/Q and finally a cross-domain validation method. The next course enters nonlinear circuit elements with AE-00 — diodes & nonlinear circuits.